000 | 07620cam a2200805 a 4500 | ||
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001 | ocn624365972 | ||
003 | OCoLC | ||
005 | 20220713030747.0 | ||
006 | m o d | ||
007 | cr cuu|||uu||| | ||
008 | 100520s2010 enka ob 001 0 eng d | ||
040 |
_aLLB _beng _epn _cLLB _dN$T _dOCLCQ _dCDX _dOCLCQ _dYDXCP _dOCLCF _dOCLCO _dOCLCQ _dAGLDB _dOCLCQ _dVTS _dVT2 _dOCLCQ _dWYU _dOCLCO _dSTF _dOCLCO _dJBG _dLEAUB _dM8D _dAJS _dOCLCO _dOCLCQ |
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_a015369161 _2Uk |
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019 |
_a666256886 _a670430586 _a1055403219 _a1063812033 _a1086435527 |
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020 |
_a9781848164468 _q(electronic bk.) |
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020 |
_a1848164467 _q(electronic bk.) |
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020 | _a9786612760006 | ||
020 | _a6612760001 | ||
020 | _z1848164459 | ||
020 | _z9781848164451 | ||
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_a(OCoLC)624365972 _z(OCoLC)666256886 _z(OCoLC)670430586 _z(OCoLC)1055403219 _z(OCoLC)1063812033 _z(OCoLC)1086435527 |
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_a276000 _bMIL |
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_aQA402.5 _b.L37 2010eb |
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072 | 7 |
_aMAT _x042000 _2bisacsh |
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082 | 0 | 4 |
_a519.6 _222 |
049 | _aMAIN | ||
100 | 1 |
_aLasserre, Jean-Bernard, _d1953- _9381471 |
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245 | 1 | 0 |
_aMoments, positive polynomials and their applications / _cJean Bernard Lasserre. |
260 |
_aLondon : _bImperial College Press ; _aSingapore : _bDistributed by World Scientific Pub. Co., _c©2010. |
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300 |
_a1 online resource (xxi, 361 pages) : _billustrations (some color). |
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336 |
_atext _btxt _2rdacontent |
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337 |
_acomputer _bc _2rdamedia |
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338 |
_aonline resource _bcr _2rdacarrier |
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490 | 1 |
_aImperial College Press optimization series, _x2041-1677 ; _vv. 1 |
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504 | _aIncludes bibliographical references (pages 341-358) and index. | ||
505 | 0 | _a1. The generalized moment problem. 1.1. Formulations. 1.2. Duality theory. 1.3. Computational complexity. 1.4. Summary. 1.5. Exercises. 1.6. Notes and sources -- 2. Positive polynomials. 2.1. Sum of squares representations and semi-definite optimization. 2.2. Nonnegative versus s.o.s. polynomials. 2.3. Representation theorems : univariate case. 2.4. Representation theorems : mutivariate case. 2.5. Polynomials positive on a compact basic semi-algebraic set. 2.6. Polynomials nonnegative on real varieties. 2.7. Representations with sparsity properties. 2.8. Representation of convex polynomials. 2.9. Summary. 2.10. Exercises. 2.11. Notes and sources -- 3. Moments. 3.1. The one-dimensional moment problem. 3.2. The multi-dimensional moment problem. 3.3. The K-moment problem. 3.4. Moment conditions for bounded density. 3.5. Summary. 3.6. Exercises. 3.7. Notes and sources -- 4. Algorithms for moment problems. 4.1. The overall approach. 4.2. Semidefinite relaxations. 4.3. Extraction of solutions. 4.4. Linear relaxations. 4.5. Extensions. 4.6. Exploiting sparsity. 4.7. Summary. 4.8. Exercises. 4.9. Notes and sources. 4.10. Proofs -- 5. Global optimization over polynomials. 5.1. The primal and dual perspectives. 5.2. Unconstrained polynomial optimization. 5.3. Constrained polynomial optimization : semidefinite relaxations. 5.4. Linear programming relaxations. 5.5. Global optimality conditions. 5.6. Convex polynomial programs. 5.7. Discrete optimization. 5.8. Global minimization of a rational function. 5.9. Exploiting symmetry. 5.10. Summary. 5.11. Exercises. 5.12. Notes and sources -- 6. Systems of polynomial equations. 6.1. Introduction. 6.2. Finding a real solution to systems of polynomial equations. 6.3. Finding all complex and/or all real solutions : a unified treatment. 6.4. Summary. 6.5. Exercises. 6.6. Notes and sources -- 7. Applications in probability. 7.1. Upper bounds on measures with moment conditions. 7.2. Measuring basic semi-algebraic sets. 7.3. Measures with given marginals. 7.4. Summary. 7.5. Exercises. 7.6. Notes and sources -- 8. Markov chains applications. 8.1. Bounds on invariant measures. 8.2. Evaluation of ergodic criteria. 8.3. Summary. 8.4. Exercises. 8.5. Notes and sources -- 9. Application in mathematical finance. 9.1. Option pricing with moment information. 9.2. Option pricing with a dynamic model. 9.3. Summary. 9.4. Notes and sources -- 10. Application in control. 10.1. Introduction. 10.2. Weak formulation of optimal control problems. 10.3. Semidefinite relaxations for the OCP. 10.4. Summary. 10.5. Notes and sources -- 11. Convex envelope and representation of convex sets. 11.1. The convex envelope of a rational function. 11.2. Semidefinite representation of convex sets. 11.3. Algebraic certificates of convexity. 11.4. Summary. 11.5. Exercises. 11.6. Notes and sources -- 12. Multivariate integration 12.1. Integration of a rational function. 12.2. Integration of exponentials of polynomials. 12.3. Maximum entropy estimation. 12.4. Summary. 12.5. Exercises. 12.6. Notes and sources -- 13. Min-max problems and Nash equilibria. 13.1. Robust polynomial optimization. 13.2. Minimizing the sup of finitely many rational cunctions. 13.3. Application to Nash equilibria. 13.4. Exercises. 13.5. Notes and sources -- 14. Bounds on linear PDE. 14.1. Linear partial differential equations. 14.2. Notes and sources. | |
520 | _aMany important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP). This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones, standard duality in convex optimization nicely expresses the duality between moments and positive polynomials. In the second part, the methodology is particularized and described in detail for various applications, including global optimization, probability, optimal control, mathematical finance, multivariate integration, etc., and examples are provided for each particular application. | ||
588 | 0 | _aPrint version record. | |
590 |
_aeBooks on EBSCOhost _bEBSCO eBook Subscription Academic Collection - Worldwide |
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650 | 0 | _aMathematical optimization. | |
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_aMoment problems (Mathematics) _9226483 |
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_aGeometry, Algebraic. _9101299 |
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_aPolynomials. _9128196 |
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_aOptimisation mathématique. _9878326 |
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_aProblèmes des moments (Mathématiques) _91463471 |
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_aGéométrie algébrique. _9894815 |
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_aPolynômes. _9951770 |
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_aMATHEMATICS _xOptimization. _2bisacsh _9924403 |
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_aGeometry, Algebraic. _2fast _0(OCoLC)fst00940902 _9101299 |
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_aMathematical optimization. _2fast _0(OCoLC)fst01012099 |
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_aMoment problems (Mathematics) _2fast _0(OCoLC)fst01025012 _9226483 |
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_aPolynomials. _2fast _0(OCoLC)fst01070715 _9128196 |
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_aPositives Polynom _2gnd _91463472 |
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655 | 0 | _aElectronic book. | |
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_aWorld Scientific (Firm) _974845 |
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_aImperial College Press optimization series ; _vv. 1. _9340731 |
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